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### 19 Cards in this Set

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 The ________ of a matrix is formed by writing its columna as rows. transpose examples: |2| A =|8|, A^T =[2 8] A matrix that has only one column is called a _____ matrix or ______ vector. column, column [ 2 ] A = | 3 | [ 2 ] FINDING THE INVERSE OF A MATRIX BY GAUSS-JORDAN ELIMINATION Let A be a square matrix of order n. 1. Write the n*2n matrix that consists of the given matrix A on the left and the n*n identity matrix I on the right to obtain [A : I]. Note that you separate the matrices A and I by a dotted line. This process is called adjoining the matrices A and I. 2. If possible, row reduce A to I using ERO's on the entire matrix [A : I]. The result will be the matrix [I : A^-1]. If this is not possible, then A is not invertible. 3. Check your work by multiplying to see that AA^-1 = I = A^-1*A. Two matrices A = [aij] and B = [bij] are ________ if they have the same size (m * n) and aij = bij for 1 <= i <= m and 1 <= j <= n. equal Properties of Zero Matrices: If A is an m*n matrix and c is a scalar, then the following properties are true. 1. A + Omn = ? 2. A + (-A) = ? 3. if cA = Omn,, then c = ? or A = ? 1. = A 2. = Omn 3. = 0 or = Omn A matrix that has only one row is called a ______ matrix or ______ vector. row, row A = [2 3 4] If A is an invertible matrix, then its inverse is ______. The inverse of A is denoted by A^-1. unique If A = [aij] and B = [bij] are matrices of size m*n, then their _____ is the m*n matrix given by A + B = [aij + bij] sum, The sum of two matrices of different sizes is undefined. |-1 2| + | 1 3| = |-1+1 2+3| = | 0 5| | 0 1| |-1 2| | 0-1 1+2| |-1 3| Find the product AB, where |-1 3| |-3 2| A = | 4 -2| and B =|-4 1| | 5 0| |-9 1| |-4 6| |-15 10| Real numbers are referred to as ______. scalars A matrix that does not have an inverse is called ______ (or _______). noninvertible singular You can use -A to represent the scalar product ______. (-1)A A - B = A + (-B) Properties of Matrix Multiplication If A, B, and C are matrices (with sizes such that the given matrix products are defined) and c is a scalar, then the following properties are true. 1. A(BC) = ? 2. A(B + C) = ? 3. (A + B)C = ? 4. c(AB) = ? 1. (AB)C (associative property of multiplication) 2. AB + AC (distributive property) 3. AC + BC (distributive property) 4. (cA)B = A(cB) If A = [aij] is an m*n matrix and B = [bij] is an n*p matrix, then the product AB is an ________ matrix. m*p AB = [cij] where cij = (Summation from k = 1 to n) aik*bkj = ai1*b1j + ai2*b2j +...+ ain*bnj An n*n matrix A is ______ (or _______) if there exist an n*n matrix B such that AB = BA = In where (In) is the identity matrix of order n. The matrix B is called the (multiplicative) _______ of A. invertible nonsingular inverse If A = [aij] is an m*n matrix and c is a scalar, then the _________ _______ of A by c is the m*n matrix given by cA = [caij]. scalar multiple Properties of the Identity Matrix If A is a matrix of size m*n, then the following properties are true. 1. AIn = ? 2. ImA = ? 1. = A 2. = A Properties of Transpose If A and B are matrices (with sizes such that the given matrix operations are defined) and c is a scalar, then the following properties are true. 1. (A^T)^T = ? 2. (A + B)^T = ? 3. (cA)^T = ? 4. (AB)^T = ? 1. A 2. A^T + B^T (transpose of a sum) 3. c(A^T) (transpose of a scalar multiple) 4. B^T*A^T (transpose of a product) Properties of matrix addition and scalar multiplication: 1. A + B = ? 2. A + (B + C) = ? 3. (cd)A = ? 4. 1A = ? 5. c(A + B) = ? 6. (c + d)A = ? 1. = B + A (commutative property of addition) 2. = (A + B) + C (associative property of addition) 3. = c(dA) 4. = A 5. cA + cB (distributive property) 6. cA + dA (distributive property)